In Madore & Freedman (1991)
we published fiducial PL relations in
seven bandpasses: *BVRIJHK*. These were all based on selecting
self-consistent sets of previously published LMC Cepheid data, scaled
to an LMC true distance modulus of 18.50 mag and applying
a single line-of-sight reddening correction using *E (B-V)* = 0.10 mag.
Thirty-two stars were available for a calibration of *BVRI* PL
relations; 25 stars were used for an alternative set of *BVRIJHK*
calibrations. In the following we compare those multiwavelength PL
relations with the Hipparcos sample of Galactic Cepheids, individually
corrected for foreground reddening and scaled to their geometric
parallax distances.
We have collected from the literature multiwavelength (*BVIJHK*) mean
magnitudes for as many of the Hipparcos-calibrating Cepheids as have
been published (notably for the infrared
Wisniewski & Johnson 1968,
Welch *et al.* 1984,
Laney & Stobie 1992
and reference therein).
These form rather disjoint subsets. After eliminating the suspected
overtone pulsators listed by
FC97,
the total available sample with
parallaxes drops from 26 to 20. Of these only 7 have mean magnitudes
published at all six wavelengths, while 10 and 13 Cepheids,
respectively have either *BVIJK* or *BVJK* magnitudes in
common. We have analyzed these four groups of stars
independently, but self-consistently, in the following way.

Using the Hipparcos parallaxes and Galactic reddenings adopted by
FC97
from
Fernie, Kamper & Seager
(1993)
scaled to the various wavelengths
using the extinction law of
Cardelli *et al.*
(1989),
we derived absolute magnitudes for each of the Cepheids in each of the
observed wavelengths. (We note that these corrections for
interstellar extinction are not inconsiderable, ranging up to 2 mag in
*B* for several stars). The resulting PL relations are shown in the
six panels of Figure 27. Error bars are
one-sigma uncertainties from
the quoted parallaxes. Note the highly correlated nature of the
individual data points about the fiducial lines. And too, we remind
the reader that the computation of distances and their related errors
from observed parallaxes is non-trivial
(Brown *et al.* 1997),
as distances are not linearly related to parallaxes, and parallax errors
can subtly bias samples. A full treatment of this issue is beyond the
scope of this paper, but we note that selection biases at least are
minimized for stars having the smallest reported errors. As discussed
by Brown *et al.*, given the true parallax distribution the
expected biases follow naturally; however, corrections to the
*observed* parallaxes require assumptions about the true distribution,
and detailed modeling. Fortunately for this application the Cepheid
sample is not parallax-selected; the objects being chosen in advance
based on their optical variability, periods and apparent magnitudes.

**Figure 27.** Multiwavelength
Period-Luminosity relations for Cepheids with Hipparcos parallaxes,
plotting all stars that have data available at the particular
wavelength, as noted in the upper left corner of each panel. In each
panel the solid sloping line is not a fit to the data, but rather it is
the published calibration of
Madore & Freedman (1991)
flanked by thin
parallel lines representing the 2- limits quoted by them as being the intrinsic width of
the instability strip at each wavelength.

The differences between these individual (trigonometric) absolute
magnitudes and the predicted *BVIJHK* magnitudes derived from the mean
PL relations of
MF91
(solid lines in Figure
27) are each plotted in
Figure 28 against the corresponding B-band
residual. The *(B-V)*
intrinsic color residuals are plotted against the B-band residuals in
the upper right panel. The individual residuals at a given wavelength
contain random contributions from the parallax uncertainties,
reddening errors, and finally the intrinsic (temperature-induced)
magnitude residuals which reflect the finite width of the Cepheid
instability strip. The observed residuals are however extremely large
(nearly 5 mag peak-to-peak) and are almost certainly dominated by the
(achromatic) errors in the parallaxes, given the strict unit-slope
correlation of the mag-mag residuals, and the total lack of any
correlation between the magnitude-color residuals
(Figure 28).

**Figure 28.** B-band residuals from the
multiwavelength Period-Luminosity relations in
Figure 27 are
sequentially plotted as a function of residuals from each of the other
five PL relations and (upper right panel) against the (B-V) color
residuals. The total lack of correlation in the latter instance is
unexpected except in the limit where the residuals are dominated by
distance errors in the derived parallaxes. This latter situation is
apparently the case given the strong (unit-slope) correlations of the
residuals in each of the other panels, regardless of wavelength.

Wavelength-dependent offsets between the six mean solutions
independently will reflect **(1)** errors in the adopted true distance to
the LMC (which set all of the zero points in the
MF91
multiwavelength
PL relation calibrations), **(2)** reddening errors in the adopted
extinction to the LMC sample of calibrating Cepheids, and finally
**(3)**
intrinsic differences between the LMC and Galactic Cepheids, for
example, due to metallicity.

Our first solution considers the largest data set (in terms of
parallaxes) but the one that is most restricted in terms of wavelength
coverage: it consists of 19 Cepheids observed in *B* and *V*.
Weighted by the square of the signal-to-noise ratio in the Hipparcos
parallax, the residuals were summed and averaged at each of the two
wavelengths giving mean offsets between the LMC calibration and the
Galactic Cepheids. The variance in each mean offset was then
calculated from the average of the squares of these same residuals
again inversely weighted by the variance in the individually quoted
parallaxes. The differences are * B* = +0.23 ± 0.35 mag and
* V* = +0.16 ±
0.28 mag, in the sense that the LMC Cepheid
calibration appears to be too faint with respect to the Galactic
calibration. (Further restricting the sample to only those 12 stars
with / _{} > 2.0 changes * B* to +0.22 ± 0.24
mag and *
V*

If the (statistically marginal, but apparently systematic) differences
in the *B* and *V* solutions were to be ascribed to reddening alone,
then the Galactic data and the LMC calibration can be reconciled by
invoking an increase of *E (B-V)* = 0.07 mag in the adopted mean
reddening to the LMC Cepheid sample. This is consistent with a
similar
suggestion regarding the LMC Cepheid calibration made recently by
Bohm-Vitense (1997)
based on different data. This reddening solution
has the consequence that it would also require the distance modulus of
the LMC to be revised *downwards* by -0.06 mag to
18.44 mag; the
uncertainty on this offset being at least as large as the uncertainty
in the individual moduli (± 0.3 mag), depending on the degree of
correlation in those cumulative uncertainties. This particular path,
of a reddening solution, cannot be considered definitive. Other
possibilities are: **(1)** the LMC true modulus should be increased by
(0.23 + 0.16) / 2 = +0.20 mag, without any change to the foreground
reddening, or **(2)** that there are differential metallicity corrections
amounting to -0.23 and -0.16 mag that need to be applied at the
*B* and *V* wavelengths, respectively. Of course, any suitably
contrived linear combination of the above three effects could also be
invoked. More constraints on the problem are obviously needed.

An alternative possibility is that some of the wavelength-dependent
effects seen in the comparison of Galactic (high metallicity) data
with the LMC (lower metallicity) data could be due to
chemical composition differences between the two samples. Taken at face value
the dependence of the apparent *V* modulus on metallicity would be
very large, *V* /
[*Fe / H*] =
0.16 / 0.15 = 1.1 (± 1.9) mag/dex, assuming that the full offset in
*V* noted in the above comparison is due to metallicity, and
adopting a metallicity underabundance of 1.4x between the LMC and the
Solar neighborhood (see, for example,
FW87).
However, we note that this
effect is basically indistinguishable from reddening in its form (as
evidenced by our first set of solutions), and that the offset
(whatever its origin) when treated as reddening leads to a true
distance modulus for the LMC that is unchanged, from previous
assumptions, at 18.50 mag. Given this apparent degeneracy between
reddening and metallicity, and the current large uncertainties in the
parallaxes, assessing the dependence on metallicity from these data
alone will remain problematic.

To obtain added leverage on the solution, moving to the infrared has
numerous well known advantages, as first articulated in
McGonegal *et al.*
(1982):
reddening effects are known to
decrease with wavelength, in a well defined and calibrated manner; and
simultaneously, metallicity effects are also expected to decrease in
amplitude with increased wavelength.

Our second solution is based on 13 Cepheids each having *BVJK* data in
common. This four-color solution gives a derived reddening
*increase* for the LMC Cepheid sample of +0.04 ± 0.08 mag, with no
formal offset in the derived 18.50 ± 0.13 mag true modulus for the
LMC. Our next approximation employs 10 Cepheids each
now having
*BVIJK* mean magnitudes. Here the formal solution for the true modulus
for the LMC is 18.53 ± 0.14 mag, with a corresponding
increase in
the mean reddening of +0.06 ± 0.07 mag. Finally, we have analyzed
a sample of 7 Galactic Cepheids, each having *BVIJHK* photometry, to
obtain one last solution: *E (B-V)* = 0.07 ± 0.07 mag with
*(m-M) _{LMC}* = 18.57 ± 0.11 mag. The fit to
this
final set of observations is shown in Figure
29; the

**Figure 29.** Apparent modulus plots for
LMC Cepheids observed at *BVIJHK* scaled to the
Hipparcos zero
point and using the published multiwavelength PL solutions of
Madore & Freedman (1991).
The solid line is a weighted ^{2} fit of a reddening
line to the data; the broken line indicates the one-sigma limits on that
solution. Inset (top left) shows the ^{2} surface indicating the minimization solution for
the modulus and reddening and the interdependence of their associated
errors.

No.Stars | µ ±
_{B} | µ ±
_{V} | µ ±
_{I} | µ ±
_{J} | µ ±
_{H} | µ ±
_{K} |
---|---|---|---|---|---|---|

19 | 18.73 ± 0.35 | 18.66 ± 0.28 | . . . | . . . | . . . | . . . |

13 | 18.71 ± 0.36 | 18.64 ± 0.24 | . . . | 18.44 ± 0.23 | . . . | 18.54 ± 0.13 |

10 | 18.74 ± 0.36 | 18.67 ± 0.24 | 18.71 ± 0.20 | 18.44 ± 0.24 | . . . | 18.57 ± 0.14 |

7 | 18.86 ± 0.36 | 18.74 ± 0.24 | 18.77 ± 0.24 | 18.62 ± 0.18 | 18.60 ± 0.15 | 18.59 ± 0.15 |

Finally, if we now adopt the metallicity correction of *V* = 0.04 mag advocated
by FC97
and assume that the effects at JHK are negligible, (and
eliminate *B* and *I* from the solution given that
metallicity corrections for these filters are not well defined at this
time) we find for this 4-color solution *E (B-V)* = 0.06 ±
0.11 mag with *(m-M) _{LMC}* = 18.57 ± 0.11 mag. This is
virtually indistinguishable from the full